qua.ostat {lmomco} | R Documentation

## Compute the Quantiles of the Distribution of an Order Statistic

### Description

This function computes a specified quantile by nonexceedance probability F for the jth-order statistic of a sample of size n for a given distribution. Let the quantile function (inverse distribution) of the Beta distribution be

mathrm{B}^{-1}(F,j,n-j+1) mbox{,}

noindent and let x(F,Theta) represent the quantile function of the given distribution and Theta represents a vector of distribution parameters. The quantile function of the distribution of the jth-order statistic is

x(mathrm{B}^{-1}(F,j,n-j+1),Theta) mbox{.}

### Usage

    qua.ostat(f,j,n,para=NULL)

### Arguments

`f` | The nonexceedance probability F for the quantile.
`j` | The jth-order statistic x_{1:n} <= x_{2:n} <= ... <= x_{j:n} <= x_{n:n}.
`n` | The sample size.
`para` | A distribution parameter list from a function such as `vec2par` or `lmom2par`.

### Value

The quantile of the distribution of the jth-order statistic is returned.

### Author(s)

W.H. Asquith

### References

Gilchrist, W.G., 2000, Statistical modelling with quantile functions: Chapman and Hall/CRC, Boca Raton, Fla.

### See Also

`lmom2par`, `vec2par`

### Examples

    gpa <- vec2par(c(100,500,0.5),type='gpa')
    n <- 20   # the sample size
    j <- 15   # the 15th order statistic
    F <- 0.99 # the 99th percentile
    theoOstat <- qua.ostat(F,j,n,gpa)

    # Let us test this value against a brute force estimate.
    Jth <- vector(mode = "numeric")
    for(i in seq(1,10000)) {
      Q <- sort(rlmomco(n,gpa))
      Jth[i] <- Q[j]
    }
    bruteOstat <- quantile(Jth,F) # estimate by built-in function
    theoOstat <- signif(theoOstat,digits=5)
    bruteOstat <- signif(bruteOstat,digits=5)
    cat(c("Theoretical=",theoOstat,"  Simulated=",bruteOstat,"\n"))

[Package lmomco version 0.92 Index]